The first part is clear from previous discussions. For any \({\bf T} = (\mathfrak {T}, \mathcal {T}) \in \mathrm {Hom}(\mathfrak {g}, \mathfrak {g}) \oplus \mathrm {Hom}(\mathcal {V}, \mathcal {V})\) , by expanding (REF ) similar to [1]}, we get that
\(\llbracket {\bf T}, {\bf T} \rrbracket _1 (x, y) =~& 2 \big ( \mathfrak {T} ([\mathfrak {T}(x), y] + [x, \mathfrak {T}(y)]) - [\mathfrak {T}(x), \mathfrak {T}(y)] \big ),\\\llbracket {\bf T}, {\bf T} \rrbracket _2 (x, u) =~& 2 \big ( \mathcal {T} ( \rho (\mathfrak {T} x) u + \rho (x) (\mathcal {T} u) ) - \rho (\mathfrak {T}x)(\mathcal {T}u) \big ),\) 
